nLab cancellative midpoint algebra

Contents

Context

Algebra

higher algebra

universal algebra

Contents

Idea

The idea of a cancellative midpoint algebra comes from Peter Freyd.

Definition

A cancellative midpoint algebra is a midpoint algebra $(M,\vert)$ that satisfies the cancellative property:

• for all $a$, $b$, and $c$ in $M$, if $a \vert b = a \vert c$, then $b = c$

Examples

The rational numbers, real numbers, and the complex numbers with $a \vert b \coloneqq \frac{a + b}{2}$ are examples of cancellative midpoint algebras.

The trivial group with $a \vert b = a \cdot b$ is a cancellative midpoint algebra.

References

• Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

Last revised on June 1, 2021 at 14:30:09. See the history of this page for a list of all contributions to it.